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Properties

Label 760.2.a.j.1.3
Level $760$
Weight $2$
Character 760.1
Self dual yes
Analytic conductor $6.069$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [760,2,Mod(1,760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("760.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-1,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 760.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.32340 q^{3} +1.00000 q^{5} -1.39821 q^{7} +2.39821 q^{9} +4.32340 q^{13} +2.32340 q^{15} +0.601793 q^{17} +1.00000 q^{19} -3.24860 q^{21} +6.04502 q^{23} +1.00000 q^{25} -1.39821 q^{27} +4.60179 q^{29} +2.79641 q^{31} -1.39821 q^{35} +1.07480 q^{37} +10.0450 q^{39} -5.44322 q^{41} -8.64681 q^{43} +2.39821 q^{45} -1.85039 q^{47} -5.04502 q^{49} +1.39821 q^{51} -3.11982 q^{53} +2.32340 q^{57} -6.69182 q^{59} -2.64681 q^{61} -3.35319 q^{63} +4.32340 q^{65} -14.4134 q^{67} +14.0450 q^{69} -5.59283 q^{71} +12.6918 q^{73} +2.32340 q^{75} -4.64681 q^{79} -10.4432 q^{81} -1.20359 q^{83} +0.601793 q^{85} +10.6918 q^{87} +9.44322 q^{89} -6.04502 q^{91} +6.49720 q^{93} +1.00000 q^{95} +4.51803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{5} - q^{7} + 4 q^{9} + 5 q^{13} - q^{15} + 5 q^{17} + 3 q^{19} + 3 q^{21} - q^{23} + 3 q^{25} - q^{27} + 17 q^{29} + 2 q^{31} - q^{35} + 8 q^{37} + 11 q^{39} + 6 q^{41} - 10 q^{43}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.32340 1.34142 0.670709 − 0.741721i \(-0.265990\pi\)
0.670709 + 0.741721i \(0.265990\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.39821 −0.528473 −0.264236 − 0.964458i \(-0.585120\pi\)
−0.264236 + 0.964458i \(0.585120\pi\)
\(8\) 0 0
\(9\) 2.39821 0.799402
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.32340 1.19910 0.599548 − 0.800339i \(-0.295347\pi\)
0.599548 + 0.800339i \(0.295347\pi\)
\(14\) 0 0
\(15\) 2.32340 0.599900
\(16\) 0 0
\(17\) 0.601793 0.145956 0.0729781 − 0.997334i \(-0.476750\pi\)
0.0729781 + 0.997334i \(0.476750\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.24860 −0.708903
\(22\) 0 0
\(23\) 6.04502 1.26047 0.630236 − 0.776403i \(-0.282958\pi\)
0.630236 + 0.776403i \(0.282958\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.39821 −0.269085
\(28\) 0 0
\(29\) 4.60179 0.854531 0.427266 − 0.904126i \(-0.359477\pi\)
0.427266 + 0.904126i \(0.359477\pi\)
\(30\) 0 0
\(31\) 2.79641 0.502251 0.251125 − 0.967955i \(-0.419199\pi\)
0.251125 + 0.967955i \(0.419199\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.39821 −0.236340
\(36\) 0 0
\(37\) 1.07480 0.176697 0.0883483 − 0.996090i \(-0.471841\pi\)
0.0883483 + 0.996090i \(0.471841\pi\)
\(38\) 0 0
\(39\) 10.0450 1.60849
\(40\) 0 0
\(41\) −5.44322 −0.850089 −0.425044 − 0.905173i \(-0.639741\pi\)
−0.425044 + 0.905173i \(0.639741\pi\)
\(42\) 0 0
\(43\) −8.64681 −1.31863 −0.659313 − 0.751869i \(-0.729153\pi\)
−0.659313 + 0.751869i \(0.729153\pi\)
\(44\) 0 0
\(45\) 2.39821 0.357504
\(46\) 0 0
\(47\) −1.85039 −0.269908 −0.134954 − 0.990852i \(-0.543089\pi\)
−0.134954 + 0.990852i \(0.543089\pi\)
\(48\) 0 0
\(49\) −5.04502 −0.720717
\(50\) 0 0
\(51\) 1.39821 0.195788
\(52\) 0 0
\(53\) −3.11982 −0.428540 −0.214270 − 0.976774i \(-0.568737\pi\)
−0.214270 + 0.976774i \(0.568737\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.32340 0.307742
\(58\) 0 0
\(59\) −6.69182 −0.871201 −0.435601 − 0.900140i \(-0.643464\pi\)
−0.435601 + 0.900140i \(0.643464\pi\)
\(60\) 0 0
\(61\) −2.64681 −0.338889 −0.169445 − 0.985540i \(-0.554197\pi\)
−0.169445 + 0.985540i \(0.554197\pi\)
\(62\) 0 0
\(63\) −3.35319 −0.422462
\(64\) 0 0
\(65\) 4.32340 0.536252
\(66\) 0 0
\(67\) −14.4134 −1.76088 −0.880441 − 0.474156i \(-0.842753\pi\)
−0.880441 + 0.474156i \(0.842753\pi\)
\(68\) 0 0
\(69\) 14.0450 1.69082
\(70\) 0 0
\(71\) −5.59283 −0.663747 −0.331873 − 0.943324i \(-0.607681\pi\)
−0.331873 + 0.943324i \(0.607681\pi\)
\(72\) 0 0
\(73\) 12.6918 1.48547 0.742733 − 0.669588i \(-0.233529\pi\)
0.742733 + 0.669588i \(0.233529\pi\)
\(74\) 0 0
\(75\) 2.32340 0.268284
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.64681 −0.522807 −0.261403 − 0.965230i \(-0.584185\pi\)
−0.261403 + 0.965230i \(0.584185\pi\)
\(80\) 0 0
\(81\) −10.4432 −1.16036
\(82\) 0 0
\(83\) −1.20359 −0.132111 −0.0660553 − 0.997816i \(-0.521041\pi\)
−0.0660553 + 0.997816i \(0.521041\pi\)
\(84\) 0 0
\(85\) 0.601793 0.0652736
\(86\) 0 0
\(87\) 10.6918 1.14628
\(88\) 0 0
\(89\) 9.44322 1.00098 0.500490 − 0.865742i \(-0.333153\pi\)
0.500490 + 0.865742i \(0.333153\pi\)
\(90\) 0 0
\(91\) −6.04502 −0.633690
\(92\) 0 0
\(93\) 6.49720 0.673728
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 4.51803 0.458736 0.229368 − 0.973340i \(-0.426334\pi\)
0.229368 + 0.973340i \(0.426334\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 760.2.a.j.1.3 ✓ 3
3.2 odd 2 6840.2.a.bg.1.2 3
4.3 odd 2 1520.2.a.s.1.1 3
5.2 odd 4 3800.2.d.l.3649.2 6
5.3 odd 4 3800.2.d.l.3649.5 6
5.4 even 2 3800.2.a.x.1.1 3
8.3 odd 2 6080.2.a.bq.1.3 3
8.5 even 2 6080.2.a.bv.1.1 3
20.19 odd 2 7600.2.a.bq.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.3 ✓ 3 1.1 even 1 trivial
1520.2.a.s.1.1 3 4.3 odd 2
3800.2.a.x.1.1 3 5.4 even 2
3800.2.d.l.3649.2 6 5.2 odd 4
3800.2.d.l.3649.5 6 5.3 odd 4
6080.2.a.bq.1.3 3 8.3 odd 2
6080.2.a.bv.1.1 3 8.5 even 2
6840.2.a.bg.1.2 3 3.2 odd 2
7600.2.a.bq.1.3 3 20.19 odd 2